Since \(S\)  is connected and Lipschitz, by the Korn-Poincaré inequality and the Rellich-Kondrachov Theorem there exists a further not relabelled subsequence \(\lbrace u_{m_h}\rbrace ,\)  a sequence \(\lbrace a_h\rbrace \)  of infinitesimal rigid displacements and \(v_0\in H^1(S;\mathbb {R}^2)\)  such that \(u_{m_h} + a_h \rightarrow v_0\)  weakly in \(H^1(S;\mathbb {R}^2)\)  and a.e. in \(S.\)
By a diagonal argument we choose a further not relabelled subsequence \(\lbrace u_{m_h}\rbrace \)  and the subset \(N\)  of indices \(i\in \mathbb {N}\)  for which there exists a ball \(B_i\subset \subset E_i\)  such that the limit of \(u_{m_h}(x) + a_h(x)\)  is finite for a.e. \(x\in B_i.\) 
We claim that for any \(i\in N\)  there exists \(v_i\in H^1_\mathrm {loc}(E_i;\mathbb {R}^2)\cap GSBD^2(E_i)\)  such that \(w_h:=u_{m_h}+ a_h \rightarrow v_i\)  weakly in \(H^1_\mathrm {loc}(E_i;\mathbb {R}^2)\)  and a.e. in \(E_i\)  as \(h\rightarrow \infty .\)
To prove the claim we fix \(i\in N\)  and let \(D\subset \subset E_i\)  be an arbitrary connected open set containing \(B_i.\)  Since \(\mathrm {sdist}(\cdot ,\partial A_{m_h})\rightarrow \mathrm {sdist}(\cdot ,\partial A)\)  locally uniformly in \(\mathbb {R}^2,\)  there exists \(h_D>0\)  such that \(D\subset \subset \mathrm {Int}(A_{m_h})\)  for all \(h>h_D.\) 
Note that \(w_h\in H^1(D)\)  and
\(\sup \limits _{h>h_D} \int _D |e(w_h)|^2dx \le C:=\frac{1}{2c_3}\sup \limits _{h>h_D} \int _{A_{m_h}\cup S} x)e(u_{m_h}):e(u_{m_h})dx<+\infty ,\) 
where in the first inequality we used (REF ) and in the second (REF ).
Since \(w_h\)  has finite limit a.e. in \(B_i\subset D,\)  by [1]} there exists \(v_i^D\in H_\mathrm {loc}^1(D)\cap GSBD^2(D)\)  and a subsequence \(\lbrace w_h^D\rbrace \)  of \(\lbrace w_h\rbrace \)  such that \(w_h^D \rightarrow v_i^D\)  weakly in \(H_\mathrm {loc}^1\)  and a.e. in \(D.\)  Now choosing a sequence \(D_1\subset \subset D_2\subset \subset \ldots \subset \subset E_i\)  of connected open sets such that \(B_i\subset D_1\)  and \(E_i = \cup _j D_j\)  and using a diagonal argument we choose a (not relablled) subsequence \(\lbrace w_h\rbrace \)  and \(v_i\in H_\mathrm {loc}^1(E_i)\cap GSBD_\mathrm {loc}^2(E_i)\)  such that \(w_h\rightarrow v_i\)  weakly in \(H_\mathrm {loc}^1(E_i)\)  and a.e. in \(E_i.\)  In particular, \(e(w_h)\rightarrow e(v_i)\)  weakly in \(L_\mathrm {loc}^2(E_i)\)  and hence, by convexity and (REF )
\(\int _{D_j} |e(v_i)|^2dx \le \liminf \limits _{h\rightarrow +\infty } \int _{D_j} |e(w_h)|^2dx \le C.\) 
Hence, letting \(j\rightarrow \infty \)  we get \(v_i\in GSBD^2(E_i).\)
By the choice of indices in \(N,\)  for any \(j\in \mathbb {N}\setminus N\)  one has \(|u_{m_h}+ a_h|\rightarrow +\infty \)  a.e. in \(E_j\)  as \(h\rightarrow +\infty .\)
Finally, since \(u_{m_h} + a_h \rightarrow u\)  weakly in \(H^1_\mathrm {loc}((\cup _{i\in N} E_i)\cup S),\)  it follows that \(e(u_{m_h})=e(u_{m_h} + a_h) \rightarrow e(u)\)  weakly in \(L^2_\mathrm {loc}((\cup _{i\in N} E_i)\cup S).\)
